
We apply our result to the chain in [1]}, which gives a parallel sampler for satisfying solutions of the CNFs in the local lemma regime. But the total work of this sampler is a large polynomial. A question is to give a parallel sampler of polylog depth with total work as the sequential sampler in [1]}, which is close to linear in \(n\) , the number of variables.

The current counting approach based on non-adaptive annealing is not work efficient.
The total work is in square of that of the adaptive annealing [3]}.
A question is how much useful the polylog depth of adaptivity can be for counting via sampling.
Is it possible to achieve the work efficiency as the adaptive annealing by parallel computing of polylog depth?

In this paper, we give a universal coupling that can couple every pair of distributions up to their Jaccard similarity.
We show that this is optimal in the worst case.
A question is whether this lower bound holds to all possible values of total variation.

A major challenge:
an \(\mathsf {RNC}\)  algorithm for sampling almost uniform (not necessarily perfect) matchings in graphs of unbounded maximum degree.
It basically asks for an \(\mathsf {RNC}\)  counterpart for the Jerrum-Sinclair algorithm [4]}, which is highly unknown whether should exist.

An even bigger challenge:
an \(\mathsf {NC}\)  algorithm (with no randomness) for approximate counting in graphs of unbounded degree.
This is even open for the poly-time sequential computation.

